A theorem for relationship between the MA process and its inversion for ARMAX identification

The paper presents a theorem to show the relationship between the parameters of the Moving Average (MA) process and those of its inversed process. The theorem can be used for the parameter identification of the MA process. It is further shown in this paper that the parameter identification of autoregressive moving average with exogenous variable model (ARMAX), based on the identification of its MA part, can be easily achieved. The approach, at first, achieves the identification of the ARX part by directly using least-square estimations to find out a straightforward relationship between estimated parameters and observed data. Then, the inversed model of the MA part is identified in a similar way. Finally, the noise variance can be computed by using identified MA parameters. Numerical simulations validate the effectiveness and efficiency of the proposed approach.